On the LX-3, (and all Nak 2 head classic transports) the problem with going in to record is almost always (usually) a sticky grease problem on the transport section that latches the Bowden cable to actuate the record switch. The fix is relatively easy if you know exactly what to do, but hopelessly difficult to clearly explain. Basically, when put in to record, the transport cam wheel reverse direction first, to latch the mechanism, so that when the transport rotates forward to engage the play/record head, the switch is activated to put the dolby and amps as inputs to the head, instead of outputs from the head. It rotates in reverse only for a second, then reveres to go forward. if the transport doesn't latch in that time period, it will not enter record.
The front panel faceplate must be removed, and the "C" spring located. Its a flat brass spring, curved with both ends looped to pivot on nylon posts. It's about at 2:00 if looking strait at the transport. The grease on the brass pivots gets sticky and prevents the spring from forcing the follower to latch and allow the switch actuation. If you remove the spring, clean it, and regrease with a light lithium, the issue usually is fixed. NOW, this ASSUMES, the long switch is not seized or oxidized. This is easily tested with the bottom cover removed and the deck put in to play, while a source input is present. If you manually actuate the switch, you will hear the sound change from tape to the source input.